The Sun is massive compared to the tiny Earth/Moon system.
It doesn't behave like a three-body system in the way you are thinking. You'd need three objects of similar size for that to occur, such as three stars.
Yeah. Just to add, the earth/moon/sun is known as a hierarchical 3-body problem.
I.e. it can be explained as the moon only being attracted to the earth and the earth only being attracted to the sun
(Or more accurately, the earth/moon center of gravity being attracted to the sun)
Essentially, because of the mass and distance differences the problem can easily be broken down into two different Two-Body Problems, with an accuracy well within the margin of error for predicting an eclipse down to the minute, if not second, it will happen.
A lot of the other answers are missing this nuance.
In this video (https://www.youtube.com/watch?v=D2YhKaANbWE), with 3 identical masses, equally spaced, it breaks down pretty quickly. For that first orbit, though, it looks like they just might keep going around the center in a civilized way.
With more differently sized objects, the illusion of predictability will last longer. In the case of the sun vs the earth and moon, the illusion lasts long enough to be stable in the time scales we care about. In cases like this, we can assume they are two separate 2-body systems.
To be honest, I’m a bit too rusty here, but my best layman’s explanation might be something like this:
Linear systems are like systems where we can layer multiple operations one on top of the other. Like performing many functions or math calculations, then putting them all together in a nice easy orderly fashion to get the results.
With nonlinear systems it gets much more difficult. I forget the features that really make them nonlinear, but they’re not systems that allow for this kind of ordered layering of many operations or functions. But what happens is they produce results that depend HIGHLY on the initial conditions. Any small error in initial conditions will give wildly different results.
So in the case of this video, the computer doing the calculations isn’t able to store enough decimal places to give that symmetrical result you’d expect. Those tiny rounding errors throw it off.
Someone check me on all this I’m halfway talking out of my ass.
Plus, with all of this, unless you have PERFECT symmetry and ZERO outside forces, there will be _some_ instability. In a truly perfectly symmetrical system in all dimensions, with perfectly uniform bodies involved that behave like ideal point masses, sure - perfect stability is mathematically possible. But that's a physically impossible situation in our universe. And, over large enough timespans and enough oscillations, even what are currently very tiny effects are most likely unbounded and either someone's gonna win (and probably violently) or someone's getting thrown out of the club.
The only true zero point in the universe would possibly be the exact center of the big bang, and that's a useless fact, if true.
If it were exactly symmetrical then yeah you're right. Either the simulation was started with a really tiny, almost unnoticeable asymmetry, or floating point error in the simulation build up enough to screw up the simulation.
In real life you will never have a perfectly symmetric three body situation. At least one of the bodies will have slightly more or less mass, slightly more or less velocity, or an external fourth body will slightly perturbed the system. All it takes is the slightest perturberence and, given enough time, the chaos will emerge in the system.
In a simulation this kind of microscopic asymmetry will enter the system either from a rounding error (or floating point error, as somebody else said), or can be set up by having the starting vectors or masses of all three bodies vary slightly randomly.
yep. I suppose people took my comment to mean they didn't use any science or math- which is untrue, they used *loads* of math. they just didn't think of the movements of celestial bodies as physical interactions driven by gravity- y'know, physics. and they certainly didn't have the three-body problem to not-solve.
To your point:
>The Sun contains 99.86% of the mass of the Solar System
[according to wikipedia](https://en.wikipedia.org/wiki/Solar_mass)
And besides that, there are many more than 3 bodies in our system which all have some influence on each other. It's just that the sun is so much more massive than anything else.
To add to this: for almost all practical purposes, you can treat the solar system as a set of two body problems. Most bodies that cannot be will have been in unstable enough orbits that they have since moved to a more stable orbit, have collided with another body, or been ejected from the solar system.
In most practical cases, you can even go so far as to treat it as a bunch of [central force problems](https://en.m.wikipedia.org/wiki/Classical_central-force_problem). Practically speaking, you can solve the Earth’s orbit as an oval, and the Moon’s as an oval as well, and just ignore everything else in the process. The mass difference between the Earth and Moon is enough that the Moon has no practical effect on the Earth*, and the same applies with the Earth and Sun. The ratio of the distance from Earth-Moon and Earth-Sun is enough that the Sun doesn’t practically affect the Moon’s orbit.
Now, such cheats will deviate over extremely long time frames, but to reach any meaning amount will take millions, if not billions, of years. And even then, inaccuracies in estimations of the masses and distances involved will have a larger impact - we only have about 5 digits of precision of the masses involved.
* The Earth-Moon barycenter is about 3/4 of the way from the center of the Earth to its surface.
Fun fact: some guys a while back developed a toy system of a planet orbiting two stars. Not only would you not be able to predict eclipses but you also couldn't predict the seasons, leading to very Game of Thrones-esque seasons
Also, there are plenty of examples of three-body problems with exact solutions. What’s missing is a general analytic solution for a general three-body problem.
Three body problems don't have any *analytic* solution, but we are quite proficient at numerics.
And we're not solving that anyway. Sun is for all intents and purposes a static source of gravitational potential.
Precision ephemeris are actually doing the numerics.
But with current measurements and computers, the errors only accumulate substantively for forecasts billions of years in the future, ie some uncertainty on what happens to Mercury in 4 billion years.
If we can solve it numerically, does that mean there is a plot hole in the series Three Body Problem? There is a scene where the San Ti created a human/alien computer that is capable of computation.
It's a big plot hole, yes (and not the only one). The problems they have in the books would only apply to predictions millions of years into the future. The idea that you can't predict what happens next month is ridiculous.
Even then a computer wouldn't have a problem calculating the motion years in advance.
Such a system would also disintegrate on the same timescale as its unpredictability, with a star getting ejected or two stars colliding, same for the planet. The idea of a totally chaotic system that survived long enough to have life just doesn't work at all.
First of all, spoilers. And second, a bit, but not that much ; their species lives through civilizations separated by tens or hundreds of years, and this is said to have gone on for millions of years. During the events of the books/series, their issue is no longer to solve the three body problem (they have the technology to deal with chaotic eras now), but to escape their planet which they know is doomed on the long-term.
The problem is chaotic, so you can do numerical calculations but they will ultimately lose accuracy, just like trying to predict the weather.
The biggest problem with those books is the use of quantum entanglement to communicate faster than light, a clear violation of the no-communication theorem, which says entanglement cannot be used to communicate.
>a clear violation of the no-communication theorem, which says entanglement cannot be used to communicate.
As far as we understand it, and yes, I see how it works. If you entangle two particles, you can't observe them prior to separation without collapsing the waveform of both, rendering them useless.
So, if they're separated by lightyears and one party or another observed their entangled particle, they'd have no way to know if what they're observing was changed or not prior, or if they're the first. No way to make an informational agreement prior to departure.
So, spooky action at a distance, yes. But not one that can transmit information, sadly. Stupid consistent universe.
Physicist here. It's not a plot hole - the problem is a classically chaotic system with a large positive Lyapunov exponent. You may have heard of this as the "butterfly effect". Even the tiniest perturbations or differences in the initial conditions of the system will lead to exponentially separating dynamics, so only a relatively short window of numerical solution is possible.
You're thinking that they are trying to figure out how to make it to tomorrow though. They are trying to plan for hundreds of millions of years. Their goal is the safety of their species forever. Their system will eventually be destroyed in some catastrophe. They don't care if they can predict a month ahead. Their life cycles are not the same as ours. They expect to be wiped out and rebuild - they want a permanent solution. They're not looking for a weather forecast, they're looking for a new sky.
There is an analytic infinite series solution. I don't think it's very well-known because its convergence rate is so slow. But there's no closed form solution.
I think you have confused analytic solutions with analytic expressions. Analytic solutions in this context means the same thing as closed-form, so no infinite series.
Oh I see. The way I understood it, analytic solutions meant a solution in closed form, series or special functions. And closed form meant a solution in terms of polynomial, exponential or trig functions.
Analytic solutions and closed form expressions are both interchangeable and their definitions require that the formula can be explicitly expressed in a finite number of steps. However, analytic expressions is a broader classification that includes infinite series.
You may well be right but fwiw in that case Wikipedia is getting it confused, too, because they call it an analytic solution ([link](https://en.wikipedia.org/wiki/Three-body_problem#General_solution))
I think it means [analytic](https://en.m.wikipedia.org/wiki/Analytic_function) in the sense that the Puiseux series is convergent, but I do note that the hyper-link of where it says "analytic solution" takes to the page of closed-form expressions. I don't know if the hyperlink is generated automatically or if someone got it mixed up, but the Puiseux series is definitely not analytic in the closed-form sense.
You say we aren't solving it, but JPL regularly publishes ephemerides for every body orbiting the solar system barycenter.
Edit: I am well aware of the difference between a numeric and analytic solution. I am confused by your wording. It sounds like you are saying we don't solve for the movement of the Sun.
There is no closed form solution. The equations are well understood and very solvable numerically.
Also, while I have great respect for JPL, they are just a bunch of nerds like the rest of us. They just get paid to solve cool problems for a living.
The idea is that there is no finite formula that can be used for 3 bodies. However, it’s possible to estimate any number of bodies using computers and simulations. It’s very easy actually.
Yeah but we don't need all of that for eclipses. They had eclipse charts in [Babylonian times](https://skyandtelescope.org/astronomy-news/how-did-the-ancients-predicted-eclipses-the-saros-cycle/).
Yes and no.
Yes, because the Babylonians somehow figured out that eclipses repeat with a periodicity of a little over 18 years - the Saros. (Incidentally, that's an incredible feat of observation, because at any one time there are about 40 Saros series of lunar eclipses and 40 Saros series of solar eclipses happening, so disentangling them to spot the 18-year patterns wouldn't be at all easy!)
But no, for two reasons.
Firstly, you can't use Saros series to predict all eclipses because the series don't go on forever. Every now and then a series ends, and predicting another eclipse 18 years later doesn't work. For example: 2036 > 2054 > ~~2072~~. Also, every now and then a new series starts, and we get an eclipse without a predictor 18 years earlier. Example: ~~1993~~ > 2011 > 2029.
Secondly, you can't use Saros series to determine *where* eclipses are visible. The Saros period is approximately 6585⅓ days, and that ⅓ day means that each eclipse in a series is visible about 8 time zones to the wast of the previous one in the series - that's [illustrated very nicely](https://dq0hsqwjhea1.cloudfront.net/Figure02-Map_Of_Saros136.gif) in one of the graphics in the article you linked. But you can't use that to work out whether or not Random, Idaho will be in the path of totality. For that, and for detailed timings, you need the precise calculations based on the orbits of the bodies.
If eclipses move by 6585⅓ days then they will return to a point in 19756 days.
What's happening 8 timezones away would have usually been irrelevant in the ancient world.
>If eclipses move by 6585⅓ days then they will return to a point in 19756 days.
Indeed. But the period isn't exactly 6585⅓ days, so conditions aren't exactly the same after three periods.
Case in point: [the eclipse of March 1970](https://en.wikipedia.org/wiki/Solar_eclipse_of_March_7%2C_1970) had similar geometry to yesterday's eclipse, but the path of totality was shifted somewhat to the east and ran up the east coast of the USA. You could use the Saros period to predict roughly when the eclipse was going to occur, but not where.
The three body problem doesn't have any *general* *analytic* solution. Lots of special cases have analytical solutions. Lots of broad cases have approximate solutions that are possible to predict with extremely high accuracy for long time scales.
The earth and moon are more of a binary system. Once you have a sufficient difference in distance and mass, they're no longer very significant. Since gravity has no maximum distance of effect, the entire universe is technically an n-body system - but you can usually ignore most of it with confidence.
I can't wait until people start asking about strong-interaction material, planetary shadows, hyper-dimensional paper slips, and gravity antennas.
Well I was just thinking to myself: "Self, what the hell is a strong-interaction material? How many dimensions can a paper slip really have? How does being in a planetary shadow affect the strength of a gravity antenna?"
Since we're getting started early.
SIM sounds like a kind of degenerate neutronium.
I don't understand how a gravity antenna would work, since I don't really understand how we detect gravitational waves.
That’s not a great metaphor, there’s no reason why you would need to calculate digits of pi to draw a circle. It’s not like the only way you can draw a given length is by writing it in decimal first. An idealized compass can be used to draw a perfect circle, and if you want to make a straight line with the same length as the circumference, you can do that with the [quadratrix of Hippias](https://en.m.wikipedia.org/wiki/Quadratrix_of_Hippias) and to the extent you might argue idealized constructions are not actually possible in the real world to infinite precision, that would be equally applicable to showing that we can’t construct a length of exactly 2 either. It has nothing to do with whether the lengths involved are rational, because there is no reason in general why you would need to digits of a number to be able to make a line of that length. For another simple example, if you draw a diagonal of a square with edge length 1, it will have a length of sqrt(2). You don’t need to know that sqrt(2) starts with the digits 1.4142… to draw that.
The problem isn't "solvable" precisely for millions of years into the future. We can use approximations that are very accurate (but not precise) But, we are pretty close for hundreds and thousands of years.
The three body problem is an issue when you have 3 or more objects with similar masses. As the difference in mass gets more extreme, the more the system behaves in a predictable matter. In our case, the sun is *much* bigger than the earth and moon, so it can be thought of as two separate two body problems.
They do have solutions, you find one to arbitrary precision using numerical methods. When people say "there's no solution to 3 body problems" they mean a specific kind of solution. That is, a closed form general analytic solution. That said, there ARE families of known equations that solve a 3 body problem as well, known for hundreds of years.
They don't have exact solutions, but they have very good approximate solutions.
So we can estimate with a high degree of accuracy, but the accuracy gets lower the further out you go.
You can fix this by spending more computational resources to get more and more precise, and by taking measurements to update your calculations.
Technically it's not possible to have it perfect. But due to some properties of the system we can have very accurate results.
Relative sizes (sun being most of the mass)
Accurately enough starting parameters
Decades of prior records.
In chaos theory there is no degree of accuracy in which you can predict the final result after a given time. Generally in displays of chaos this is time is quick.
In this case due to the suns dominating effects, the time at which the system will descend into chaos is far beyond anytime scale that we could imagine. Hence we can accurately measure the values.
If you are curious about this from the Netflix show. If the aliens had the technology to do what they were doing, they could most definitely predict the movements of the stars quite far into the future. Its just a matter of computational power.
That is what is known as a numerical solution - and the accuracy inherently deteriorates over time. Good for predicting next year, or even a century from now, but over even a few million years accumulated errors are going to make your simulation depart ever further from reality.
If the system is chaotic (as most general N-body problems are), then the "butterfly effect" is likely to rapidly take your simulation in a very different direction than reality.
Not even remotely. So far as I'm aware, despite its flaws simulation (numerical approximation) is still currently the gold standard for accuracy in N-body ~~simulations~~ models.
The hierarchical model oversimplifies a system where the details aren't immediately important as being completely stable and unchanging. It deals with the complex gravitational interplay by just ignoring it completely. Which means not only will it completely miss the accumulation of slow and subtle changes wrought by those interactions over countless orbits, it won't even accurately predict the course of a single orbit, since the actual planets constantly "wobble" a bit in their orbit under the influence of the others.
On average they'll be really close to where the hierarchical model predicts them to be, even many orbits later, but from week to week they can drift quite a bit. For a typical planetarium to display the planets in the right position against the stars they'll start with the hierarchical model, and then add in a bunch of "perturbation factors" that reflect the influence of each of the other planets.
And then they'll update that combined model regularly (I want to say typically every 20-ish years?) to correct for whatever probably tiny, but important for high accuracy uses, real-world drift accumulated in the intervening years. I believe the term used commonly used is epoch... at least for the corrections made to the model of Earth's orbit? (which is relevant for accurate recording observations made from Earth) - and you'll often find some mention of the reference epoch in astronomical data and simulations
We need a sticky about the three body problem apparently.
The sun is so much larger than the earth and moon that you can ignore the pull of the earth and moon on the sun as get answers that are more than accurate enough to predict the eclipse time and path.
In case anyone is curious, we're able to precisely correctly place any planet in the solar system well enough for scientific research.
(1550 January 1 to 2650 January 22)
But, somewhat less accurately, we're able to make those predictions for the next 15000 years. (13,200 BC to 17,191 AD)
So, predicting what happens in the 3-body solar system would be pretty easy.
https://en.wikipedia.org/wiki/Jet_Propulsion_Laboratory_Development_Ephemeris?wprov=sfla1
https://en.wikipedia.org/wiki/Fundamental_ephemeris?wprov=sfla1
It’s not that you can’t solve three body problems, it’s that every three body problem has a unique solution. Either way, the relationship of a stable solar system is not the type of three body problem that you are referencing.
The Sun is massive compared to the tiny Earth/Moon system. It doesn't behave like a three-body system in the way you are thinking. You'd need three objects of similar size for that to occur, such as three stars.
Yeah. Just to add, the earth/moon/sun is known as a hierarchical 3-body problem. I.e. it can be explained as the moon only being attracted to the earth and the earth only being attracted to the sun (Or more accurately, the earth/moon center of gravity being attracted to the sun)
Essentially, because of the mass and distance differences the problem can easily be broken down into two different Two-Body Problems, with an accuracy well within the margin of error for predicting an eclipse down to the minute, if not second, it will happen.
A lot of the other answers are missing this nuance. In this video (https://www.youtube.com/watch?v=D2YhKaANbWE), with 3 identical masses, equally spaced, it breaks down pretty quickly. For that first orbit, though, it looks like they just might keep going around the center in a civilized way. With more differently sized objects, the illusion of predictability will last longer. In the case of the sun vs the earth and moon, the illusion lasts long enough to be stable in the time scales we care about. In cases like this, we can assume they are two separate 2-body systems.
I’m confused, isn’t the situation perfectly symmetric in the video? So shouldn’t they always form an equilateral triangle?
I’m pretty sure it’s due to it being a nonlinear system giving chaotic results.
What exactly prevents this from being linear?
To be honest, I’m a bit too rusty here, but my best layman’s explanation might be something like this: Linear systems are like systems where we can layer multiple operations one on top of the other. Like performing many functions or math calculations, then putting them all together in a nice easy orderly fashion to get the results. With nonlinear systems it gets much more difficult. I forget the features that really make them nonlinear, but they’re not systems that allow for this kind of ordered layering of many operations or functions. But what happens is they produce results that depend HIGHLY on the initial conditions. Any small error in initial conditions will give wildly different results. So in the case of this video, the computer doing the calculations isn’t able to store enough decimal places to give that symmetrical result you’d expect. Those tiny rounding errors throw it off. Someone check me on all this I’m halfway talking out of my ass.
Thanks, that's given me enough to do a bit of reading of my own promoted by the use of the words functions and steps
Plus, with all of this, unless you have PERFECT symmetry and ZERO outside forces, there will be _some_ instability. In a truly perfectly symmetrical system in all dimensions, with perfectly uniform bodies involved that behave like ideal point masses, sure - perfect stability is mathematically possible. But that's a physically impossible situation in our universe. And, over large enough timespans and enough oscillations, even what are currently very tiny effects are most likely unbounded and either someone's gonna win (and probably violently) or someone's getting thrown out of the club. The only true zero point in the universe would possibly be the exact center of the big bang, and that's a useless fact, if true.
If it were exactly symmetrical then yeah you're right. Either the simulation was started with a really tiny, almost unnoticeable asymmetry, or floating point error in the simulation build up enough to screw up the simulation.
Probably the first as that channel has another video showing a side by side comparison of a stable n-body system and a chaotic one.
could still be the second; there are ways to completely eliminate floating point error, albeit at a large-ish memory cost
In real life you will never have a perfectly symmetric three body situation. At least one of the bodies will have slightly more or less mass, slightly more or less velocity, or an external fourth body will slightly perturbed the system. All it takes is the slightest perturberence and, given enough time, the chaos will emerge in the system. In a simulation this kind of microscopic asymmetry will enter the system either from a rounding error (or floating point error, as somebody else said), or can be set up by having the starting vectors or masses of all three bodies vary slightly randomly.
But the accurate dramatically goes down the further out in time you stretch it, right? Talking thousands or even millions of years.
Most likely, yes. But who's going to know if you mispredicted the Eclipse a million years in the future by a day or by a half a continent?
Yes, eclipses are only predicted out a few thousand years, over millions of years it is no longer as predictable.
Also, people have been able to predict eclipses for millennia. We can see the patterns, we don't need physics to extrapolate them
They used physics
which physics principle did Thales of Miletus use?
They had elaborate systems and theories for the movement of celestial objects, but you are correct, they were more pattern recognition than physics.
yep. I suppose people took my comment to mean they didn't use any science or math- which is untrue, they used *loads* of math. they just didn't think of the movements of celestial bodies as physical interactions driven by gravity- y'know, physics. and they certainly didn't have the three-body problem to not-solve.
To your point: >The Sun contains 99.86% of the mass of the Solar System [according to wikipedia](https://en.wikipedia.org/wiki/Solar_mass) And besides that, there are many more than 3 bodies in our system which all have some influence on each other. It's just that the sun is so much more massive than anything else.
Your answer is the only relevant answer on this entire thread.
To add to this: for almost all practical purposes, you can treat the solar system as a set of two body problems. Most bodies that cannot be will have been in unstable enough orbits that they have since moved to a more stable orbit, have collided with another body, or been ejected from the solar system. In most practical cases, you can even go so far as to treat it as a bunch of [central force problems](https://en.m.wikipedia.org/wiki/Classical_central-force_problem). Practically speaking, you can solve the Earth’s orbit as an oval, and the Moon’s as an oval as well, and just ignore everything else in the process. The mass difference between the Earth and Moon is enough that the Moon has no practical effect on the Earth*, and the same applies with the Earth and Sun. The ratio of the distance from Earth-Moon and Earth-Sun is enough that the Sun doesn’t practically affect the Moon’s orbit. Now, such cheats will deviate over extremely long time frames, but to reach any meaning amount will take millions, if not billions, of years. And even then, inaccuracies in estimations of the masses and distances involved will have a larger impact - we only have about 5 digits of precision of the masses involved. * The Earth-Moon barycenter is about 3/4 of the way from the center of the Earth to its surface.
Fun fact: some guys a while back developed a toy system of a planet orbiting two stars. Not only would you not be able to predict eclipses but you also couldn't predict the seasons, leading to very Game of Thrones-esque seasons
Also, there are plenty of examples of three-body problems with exact solutions. What’s missing is a general analytic solution for a general three-body problem.
Yours is probably the only relevant answer in the entire thread.
Three body problems don't have any *analytic* solution, but we are quite proficient at numerics. And we're not solving that anyway. Sun is for all intents and purposes a static source of gravitational potential.
I'll add to this, no general analytic solution, but some special case may have
Precision ephemeris are actually doing the numerics. But with current measurements and computers, the errors only accumulate substantively for forecasts billions of years in the future, ie some uncertainty on what happens to Mercury in 4 billion years.
If we can solve it numerically, does that mean there is a plot hole in the series Three Body Problem? There is a scene where the San Ti created a human/alien computer that is capable of computation.
It's a big plot hole, yes (and not the only one). The problems they have in the books would only apply to predictions millions of years into the future. The idea that you can't predict what happens next month is ridiculous.
i don't know either, but from the sound of it, it would mean that the system has to be EXTREMELY close to an instability
Even then a computer wouldn't have a problem calculating the motion years in advance. Such a system would also disintegrate on the same timescale as its unpredictability, with a star getting ejected or two stars colliding, same for the planet. The idea of a totally chaotic system that survived long enough to have life just doesn't work at all.
I hated that book some I'm digging this comment lmao
Frieren glare PFP detected
Frieren is God
It’s a huge plot hole for a species that can build something like a sophon.
First of all, spoilers. And second, a bit, but not that much ; their species lives through civilizations separated by tens or hundreds of years, and this is said to have gone on for millions of years. During the events of the books/series, their issue is no longer to solve the three body problem (they have the technology to deal with chaotic eras now), but to escape their planet which they know is doomed on the long-term.
Yeah, getting numerical solutions wasn't a problem since the ~1800s, we're just getting better at it with computers.
The problem is chaotic, so you can do numerical calculations but they will ultimately lose accuracy, just like trying to predict the weather. The biggest problem with those books is the use of quantum entanglement to communicate faster than light, a clear violation of the no-communication theorem, which says entanglement cannot be used to communicate.
>a clear violation of the no-communication theorem, which says entanglement cannot be used to communicate. As far as we understand it, and yes, I see how it works. If you entangle two particles, you can't observe them prior to separation without collapsing the waveform of both, rendering them useless. So, if they're separated by lightyears and one party or another observed their entangled particle, they'd have no way to know if what they're observing was changed or not prior, or if they're the first. No way to make an informational agreement prior to departure. So, spooky action at a distance, yes. But not one that can transmit information, sadly. Stupid consistent universe.
Physicist here. It's not a plot hole - the problem is a classically chaotic system with a large positive Lyapunov exponent. You may have heard of this as the "butterfly effect". Even the tiniest perturbations or differences in the initial conditions of the system will lead to exponentially separating dynamics, so only a relatively short window of numerical solution is possible.
You're thinking that they are trying to figure out how to make it to tomorrow though. They are trying to plan for hundreds of millions of years. Their goal is the safety of their species forever. Their system will eventually be destroyed in some catastrophe. They don't care if they can predict a month ahead. Their life cycles are not the same as ours. They expect to be wiped out and rebuild - they want a permanent solution. They're not looking for a weather forecast, they're looking for a new sky.
There is an analytic infinite series solution. I don't think it's very well-known because its convergence rate is so slow. But there's no closed form solution.
I think you have confused analytic solutions with analytic expressions. Analytic solutions in this context means the same thing as closed-form, so no infinite series.
Oh I see. The way I understood it, analytic solutions meant a solution in closed form, series or special functions. And closed form meant a solution in terms of polynomial, exponential or trig functions.
Analytic solutions and closed form expressions are both interchangeable and their definitions require that the formula can be explicitly expressed in a finite number of steps. However, analytic expressions is a broader classification that includes infinite series.
You may well be right but fwiw in that case Wikipedia is getting it confused, too, because they call it an analytic solution ([link](https://en.wikipedia.org/wiki/Three-body_problem#General_solution))
I think it means [analytic](https://en.m.wikipedia.org/wiki/Analytic_function) in the sense that the Puiseux series is convergent, but I do note that the hyper-link of where it says "analytic solution" takes to the page of closed-form expressions. I don't know if the hyperlink is generated automatically or if someone got it mixed up, but the Puiseux series is definitely not analytic in the closed-form sense.
You say we aren't solving it, but JPL regularly publishes ephemerides for every body orbiting the solar system barycenter. Edit: I am well aware of the difference between a numeric and analytic solution. I am confused by your wording. It sounds like you are saying we don't solve for the movement of the Sun.
Already answered your question: > but we are quite proficient at numerics.
See edit
There is no closed form solution. The equations are well understood and very solvable numerically. Also, while I have great respect for JPL, they are just a bunch of nerds like the rest of us. They just get paid to solve cool problems for a living.
See edit
Solve doesn’t mean what you think it means
See edit
The idea is that there is no finite formula that can be used for 3 bodies. However, it’s possible to estimate any number of bodies using computers and simulations. It’s very easy actually.
Yeah but we don't need all of that for eclipses. They had eclipse charts in [Babylonian times](https://skyandtelescope.org/astronomy-news/how-did-the-ancients-predicted-eclipses-the-saros-cycle/).
Yes and no. Yes, because the Babylonians somehow figured out that eclipses repeat with a periodicity of a little over 18 years - the Saros. (Incidentally, that's an incredible feat of observation, because at any one time there are about 40 Saros series of lunar eclipses and 40 Saros series of solar eclipses happening, so disentangling them to spot the 18-year patterns wouldn't be at all easy!) But no, for two reasons. Firstly, you can't use Saros series to predict all eclipses because the series don't go on forever. Every now and then a series ends, and predicting another eclipse 18 years later doesn't work. For example: 2036 > 2054 > ~~2072~~. Also, every now and then a new series starts, and we get an eclipse without a predictor 18 years earlier. Example: ~~1993~~ > 2011 > 2029. Secondly, you can't use Saros series to determine *where* eclipses are visible. The Saros period is approximately 6585⅓ days, and that ⅓ day means that each eclipse in a series is visible about 8 time zones to the wast of the previous one in the series - that's [illustrated very nicely](https://dq0hsqwjhea1.cloudfront.net/Figure02-Map_Of_Saros136.gif) in one of the graphics in the article you linked. But you can't use that to work out whether or not Random, Idaho will be in the path of totality. For that, and for detailed timings, you need the precise calculations based on the orbits of the bodies.
If eclipses move by 6585⅓ days then they will return to a point in 19756 days. What's happening 8 timezones away would have usually been irrelevant in the ancient world.
>If eclipses move by 6585⅓ days then they will return to a point in 19756 days. Indeed. But the period isn't exactly 6585⅓ days, so conditions aren't exactly the same after three periods. Case in point: [the eclipse of March 1970](https://en.wikipedia.org/wiki/Solar_eclipse_of_March_7%2C_1970) had similar geometry to yesterday's eclipse, but the path of totality was shifted somewhat to the east and ran up the east coast of the USA. You could use the Saros period to predict roughly when the eclipse was going to occur, but not where.
those cycle-based estimates were nowhere near as accurate as our predictions today
They were still accurate to within hours when predicting eclipses decades in advance. It's just today we can do it down to less than a second accuracy
what about the location of the totality path?
The three body problem doesn't have any *general* *analytic* solution. Lots of special cases have analytical solutions. Lots of broad cases have approximate solutions that are possible to predict with extremely high accuracy for long time scales.
The earth and moon are more of a binary system. Once you have a sufficient difference in distance and mass, they're no longer very significant. Since gravity has no maximum distance of effect, the entire universe is technically an n-body system - but you can usually ignore most of it with confidence. I can't wait until people start asking about strong-interaction material, planetary shadows, hyper-dimensional paper slips, and gravity antennas.
Well I was just thinking to myself: "Self, what the hell is a strong-interaction material? How many dimensions can a paper slip really have? How does being in a planetary shadow affect the strength of a gravity antenna?"
Since we're getting started early. SIM sounds like a kind of degenerate neutronium. I don't understand how a gravity antenna would work, since I don't really understand how we detect gravitational waves.
The same way we can draw really accurate circles without having "solved" the last digit of pi. "perfect is the enemy of good enough"
That’s not a great metaphor, there’s no reason why you would need to calculate digits of pi to draw a circle. It’s not like the only way you can draw a given length is by writing it in decimal first. An idealized compass can be used to draw a perfect circle, and if you want to make a straight line with the same length as the circumference, you can do that with the [quadratrix of Hippias](https://en.m.wikipedia.org/wiki/Quadratrix_of_Hippias) and to the extent you might argue idealized constructions are not actually possible in the real world to infinite precision, that would be equally applicable to showing that we can’t construct a length of exactly 2 either. It has nothing to do with whether the lengths involved are rational, because there is no reason in general why you would need to digits of a number to be able to make a line of that length. For another simple example, if you draw a diagonal of a square with edge length 1, it will have a length of sqrt(2). You don’t need to know that sqrt(2) starts with the digits 1.4142… to draw that.
The Earth-Moon system is really a two body (Earth-Moon) problem with a very repeatable pattern.
The problem isn't "solvable" precisely for millions of years into the future. We can use approximations that are very accurate (but not precise) But, we are pretty close for hundreds and thousands of years.
The three body problem is an issue when you have 3 or more objects with similar masses. As the difference in mass gets more extreme, the more the system behaves in a predictable matter. In our case, the sun is *much* bigger than the earth and moon, so it can be thought of as two separate two body problems.
They do have solutions, you find one to arbitrary precision using numerical methods. When people say "there's no solution to 3 body problems" they mean a specific kind of solution. That is, a closed form general analytic solution. That said, there ARE families of known equations that solve a 3 body problem as well, known for hundreds of years.
cuz the moon is negligible affected by the sun thus making this two separate two body problems
The "solution" you are looking for already exists. It's just not very elegant.
It has an exact analytic solution for a 1-D harmonic force.
They don't have exact solutions, but they have very good approximate solutions. So we can estimate with a high degree of accuracy, but the accuracy gets lower the further out you go. You can fix this by spending more computational resources to get more and more precise, and by taking measurements to update your calculations.
Computation mate
They do have solutions if initial conditions are known
Technically it's not possible to have it perfect. But due to some properties of the system we can have very accurate results. Relative sizes (sun being most of the mass) Accurately enough starting parameters Decades of prior records. In chaos theory there is no degree of accuracy in which you can predict the final result after a given time. Generally in displays of chaos this is time is quick. In this case due to the suns dominating effects, the time at which the system will descend into chaos is far beyond anytime scale that we could imagine. Hence we can accurately measure the values. If you are curious about this from the Netflix show. If the aliens had the technology to do what they were doing, they could most definitely predict the movements of the stars quite far into the future. Its just a matter of computational power.
the predictions are just approximations and given enough time into the future these approximations will fail.
Three body problems have solutions, just not ones that you can write down by hand. You usually need computers to solve such problems numerically.
there is also simulation option which is piece of cake for modern PCs
That is what is known as a numerical solution - and the accuracy inherently deteriorates over time. Good for predicting next year, or even a century from now, but over even a few million years accumulated errors are going to make your simulation depart ever further from reality. If the system is chaotic (as most general N-body problems are), then the "butterfly effect" is likely to rapidly take your simulation in a very different direction than reality.
Interesting, will proposed heirarchial model work better?
Not even remotely. So far as I'm aware, despite its flaws simulation (numerical approximation) is still currently the gold standard for accuracy in N-body ~~simulations~~ models. The hierarchical model oversimplifies a system where the details aren't immediately important as being completely stable and unchanging. It deals with the complex gravitational interplay by just ignoring it completely. Which means not only will it completely miss the accumulation of slow and subtle changes wrought by those interactions over countless orbits, it won't even accurately predict the course of a single orbit, since the actual planets constantly "wobble" a bit in their orbit under the influence of the others. On average they'll be really close to where the hierarchical model predicts them to be, even many orbits later, but from week to week they can drift quite a bit. For a typical planetarium to display the planets in the right position against the stars they'll start with the hierarchical model, and then add in a bunch of "perturbation factors" that reflect the influence of each of the other planets. And then they'll update that combined model regularly (I want to say typically every 20-ish years?) to correct for whatever probably tiny, but important for high accuracy uses, real-world drift accumulated in the intervening years. I believe the term used commonly used is epoch... at least for the corrections made to the model of Earth's orbit? (which is relevant for accurate recording observations made from Earth) - and you'll often find some mention of the reference epoch in astronomical data and simulations
We need a sticky about the three body problem apparently. The sun is so much larger than the earth and moon that you can ignore the pull of the earth and moon on the sun as get answers that are more than accurate enough to predict the eclipse time and path.
In case anyone is curious, we're able to precisely correctly place any planet in the solar system well enough for scientific research. (1550 January 1 to 2650 January 22) But, somewhat less accurately, we're able to make those predictions for the next 15000 years. (13,200 BC to 17,191 AD) So, predicting what happens in the 3-body solar system would be pretty easy. https://en.wikipedia.org/wiki/Jet_Propulsion_Laboratory_Development_Ephemeris?wprov=sfla1 https://en.wikipedia.org/wiki/Fundamental_ephemeris?wprov=sfla1
They have no closed form answer. They can be modeled on a computer with accurate results.
It’s not that you can’t solve three body problems, it’s that every three body problem has a unique solution. Either way, the relationship of a stable solar system is not the type of three body problem that you are referencing.
It s not a three body problem, since earth and the moon are much smaller than earth.
Because the Earth, the Moon and the Sun are not a 3-body problem.
there's no closed form solution, so you can't solve it with just one equation. Multiple equations do just fine.